Sequences, series, summability Definition

Hölder summation

Builds on Summability method Partial sum Cesàro mean

Definition

Let $\sum_{n \ge 0} a_n$ be a series of complex numbers with partial sums $s_n = a_0 + \cdots + a_n$ . Put $H^0_n = s_n$ , and for each integer $k \ge 1$ let $H^k_n$ be the Cesàro mean of the sequence $(H^{k-1}_n)$ : $H^k_n = \frac{H^{k-1}_0 + H^{k-1}_1 + \cdots + H^{k-1}_n}{n + 1}.$ The series is Hölder summable of order $k$ , or $(H, k)$ summable, to $s$ if $H^k_n \to s$ as $n \to \infty$ .

Remarks

The method $(H, 0)$ is ordinary convergence, and $(H, 1)$ is Cesàro summation of order $1$ .

Example: for $1 - 2 + 3 - 4 + \cdots$ the partial sums are $1, -1, 2, -2, 3, \dots$ . Then $H^1_{2m} = \frac{m+1}{2m+1}$ and $H^1_{2m+1} = 0$ . So $H^1_n$ is the sequence $\frac12, 0, \frac12, 0, \dots$ up to terms that tend to $0$ , and its averages $H^2_n$ tend to $\frac14$ . The series is $(H, 2)$ summable to $1/4$ .

Non-example: the same series is not $(H, 1)$ summable, since $H^1_n$ has the two limit points $1/2$ and $0$ .

By a theorem of Knopp and Schur, for each $k$ a series is $(H, k)$ summable to $s$ if and only if it is Cesàro $(C, k)$ summable to $s$ .

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Sequence definition
Summability method concept
Real sequence definition
Cesàro mean definition
Series definition
Partial sum definition
Hölder summation definition

Tag	`TA8NQQCQCF`
URI	`m2://TA8NQQCQCF`
Kind	definition
Writing	W0 stub
MSC	40G05
Defines	Hölder summation
Uses	Summability method, Partial sum, Cesàro mean
Changed	2 November 2026

Hölder summation

Definition

Remarks

Reading path

Further reading

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